It turns out that the Remez exchange algorithm has convergence
problems for filters larger than a few hundred taps. Therefore, the
FIR filter length
was chosen above to be small enough to work
out in this comparison. However, keep in mind that for very large
filter orders, the Remez exchange method may not be an option. There
are more recently developed methods for optimal Chebyshev FIR filter
design, using ``convex optimization'' techniques, that may continue to
work at very high orders
[218,22,153]. The fast nonparametric
methods discussed above (frequency sampling, window method) will work
fine at extremely high orders.

Instead, however, we will use a more robust method
[228] which uses the Remez exchange
algorithm to design a lowpass filter, followed by modulation of
the lowpass impulse-response by a complex sinusoid at frequency
in order to frequency-shift the lowpass to the single-sideband
filter we seek:

The weighting [1,10] in the call to firpm above says
``make the pass-band ripple
times that of the stop-band.'' For
steady-state audio spectra, pass-band ripple can be as high as dB or more without audible consequences.5.11 The result is
shown in Fig.4.16 (full amplitude response) and
Fig.4.17 (zoom-in on the dc transition band). By
symmetry the high-frequency transition region is identical (but
flipped):

Figure 4.16:Frequency response of
the optimal Chebyshev FIR filter designed by the Remez exchange
algorithm.